This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present technological advancement. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the technological advancement. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Controlled-source electromagnetic (“CSEM”) geophysical surveys use man-made sources to generate electromagnetic fields to excite the earth, and deploy receiver instruments on the earth's surface, on the seafloor, in the air, or inside boreholes to measure the resulting electric and magnetic fields, i.e., the earth's response to the source excitation. By way of example, a vessel can tow a submerged CSEM transmitter over an area of the seafloor. The electric and magnetic fields measured by receivers are then analyzed to determine the electrical resistivity of the earth structures beneath the seafloor.
Magnetotelluric (“MT”) geophysical surveys exploit naturally occurring variations in the earth's electromagnetic fields. Receivers are deployed on the earth's surface, on the seafloor, in the air, or inside boreholes to measure the vector components of either the electric field, or the magnetic field, or both over a length in time of this natural variation. Transfer functions between the measured fields are estimated which are then analyzed to determine electrical resistivity of the earth structure beneath the plane of measurement.
One manifestation of mathematical inversion is the process by which observations, or data, are converted into estimates of a model of interest. For example, in geophysics these observations may be electric and magnetic field measurements at the earth's surface, and the model of interest is the distribution of electrical resistivity within the subsurface. The physical process which connects electrical resistivity, the model, to the electromagnetic fields, the data, is represented by a non-linear set of equations. As such, the inversion algorithm used must be chosen from a family of non-linear optimizations. Two of the non-linear optimization styles, most commonly used in geophysics are Non-linear Conjugate Gradient (NLCG), a first-order approach, and Gauss-Newton, a second-order approach (e.g., Newman & Alumbaugh, 2000; D. Avdeev, 2005; Hu et al., 2011; Egbert, 2012). In short, the NLCG method is far less onerous computationally than Gauss-Newton, but is less adept at resolving finer features of a model than a second-order approach, like Gauss-Newton.
It is fair to say that independent of computational cost, between the two aforementioned approaches, the Gauss-Newton method is always preferred, but because of its high cost, it is seldom used for large problems like 3D inversion.
Occam's inversion algorithm was first introduced by Constable et al. (1987) to find the smoothest model that fits some MT or geo-electric sounding data. Occam's inversion is a process that matches the measured electromagnetic data to a theoretical model of many layers of finite thickness and varying resistivity. Details of conventional Occam inversion are found in Constable et al., and are only discussed herein as may be necessary for those of ordinary skill in the art to understanding the present technological advancement.
For small-scale problems, Occam's inversion is widely used in underdetermined non-linear inversion, like that encountered in electromagnetic geophysics. It is a regularized Gauss-Newton method which progresses iteratively over a series of models which tend to produce lower misfit than their predecessor. Within the iteration, a search is made for the optimal model update by varying trade-offs between the minimizing data misfit and minimizing model structure (often called regularization). Such a trade-off is set by a Lagrange parameter, λ. Each step in the search requires a large matrix inversion to find a prospective model update, which is used to find the corresponding misfit achieved by the current Lagrange parameter value. Often this “line” search requires the testing of 5-10 different Lagrange parameter values before a suitable reduction in misfit is achieved. The size of the linear system matrix, and therefore the duration for computing its inverse, is a function of the number of model parameters. As geophysical problems become more complicated, i.e. 2D or 3D inversion, the number of these model parameters can become very large, and therefore the Lagrange parameter line search, requiring multiple large matrix inversions, can be debilitating slow.